Graphical Method – Plotting Parabolas, Vertex, and Roots

1. Introduction In this lesson, you will learn how to solve quadratic equations graphically. By plotting quadratic expressions as curves on a coordinate plane, you can visualize key algebraic features, locate the vertex, find the axis of symmetry, and find the roots of the equation.

2. Core Concept Explanation Every quadratic equation y = ax^2 + bx + c forms a curve called a parabola. * Shape: If a > 0, the parabola opens upward (\cup), representing a minimum turning point. If a < 0, it opens downward (\cap), representing a maximum turning point. * Vertex (Turning Point): The peak or lowest point of the curve. The X-coordinate is given by x = -\frac{b}{2a}, and the Y-coordinate is found by substituting this X-coordinate back into the equation. * Axis of Symmetry: The vertical line that cuts the parabola into two symmetrical halves: x = -\frac{b}{2a}. * Roots (X-intercepts): The points where the curve crosses the X-axis (where y = 0).

3. Step-by-Step Method To solve a quadratic equation graphically: 1. Complete a table of values calculating y for a specified range of x (e.g., -3 \le x \le 3). 2. Draw the vertical Y-axis and horizontal X-axis on graph paper, choosing a consistent scale. 3. Plot the (x, y) coordinate points on the graph. 4. Draw a smooth, continuous freehand curve to connect the points. Do not use a straight ruler. 5. Identify the roots by reading the X-coordinates where the curve crosses the X-axis (y = 0).

4. Worked Examples * Example 1: Draw the graph of y = x^2 - 4x + 3 for 0 \le x \le 4, and find the roots. * Solution: 1. Table of values: * x=0 \Rightarrow y=3 * x=1 \Rightarrow y=0 * x=2 \Rightarrow y=-1 (vertex) * x=3 \Rightarrow y=0 * x=4 \Rightarrow y=3 2. Coordinates to plot: (0, 3), (1, 0), (2, -1), (3, 0), (4, 3). 3. The curve crosses the X-axis at x = 1 and x = 3. These are the roots. * Example 2: Find the axis of symmetry and vertex for y = -2x^2 + 8x - 5. * Solution: 1. Coefficients: a = -2, b = 8. 2. Axis of symmetry: x = -\frac{b}{2a} = -\frac{8}{2(-2)} = 2. 3. Vertex: Substitute x=2: y = -2(2)^2 + 8(2) - 5 = -8 + 16 - 5 = 3. 4. Vertex is at (2, 3). Since a < 0, it is a maximum point. * Example 3: Solve x^2 - x - 2 = 0 using a table of values for -2 \le x \le 3. * Solution: 1. Table of values: * x=-2 \Rightarrow y=4, x=-1 \Rightarrow y=0, x=0 \Rightarrow y=-2, x=1 \Rightarrow y=-2, x=2 \Rightarrow y=0, x=3 \Rightarrow y=4. 2. Plot the points and draw the curve. 3. The roots (X-intercepts) are x = -1 and x = 2.

5. Exam Relevance * Example WAEC Question: Plot the graph of y = 2x^2 - 4x - 3 for -2 \le x \le 4. Using the graph, solve 2x^2 - 4x - 3 = 0. * Solution: The coordinates are calculated as: (-2, 13), (-1, 3), (0, -3), (1, -5) [minimum], (2, -3), (3, 3), (4, 13). Plotting these coordinates and reading the x-intercepts gives the solutions: x \approx -0.6 and x \approx 2.6.

6. Common Mistakes * Using a ruler to draw the curve: Joining points with straight line segments rather than drawing a smooth freehand curve. * Incorrect scale implementation: Failing to follow the specific grid constraints given (e.g. 2 cm to 5 units).

7. Practice Questions 1. Plot y = -x^2 + 2x + 3 for -2 \le x \le 4. What is the maximum value of y? * Answer: The maximum value is y = 4 (at the vertex (1, 4)). 2. Find the roots of x^2 - 4 = 0 graphically. * Answer: x = -2 and x = 2.

8. Summary / Cheat Sheet * Parabolas open up (\cup) if a > 0, down (\cap) if a < 0. * Axis of Symmetry formula: x = -\frac{b}{2a}. * Turning point (vertex) is always located on the axis of symmetry. * Roots are where the parabola crosses the X-axis.